The Expressiveness of Number

For me, one of the more intriguing things that happened in mathematics is what is called the arithmetization of the Calculus.  This is not because it contributes to my understanding of fundamental concepts (because it doesn’t).  Nor is it because the ideas are exotic (they’re not). I’m captivated, instead, by what it may demonstrate about the way we see – the way we come to understand or discern meaning.  How does one separate calculus ideas from the intuitive notions of space and motion that gave rise to it?  That this was accomplished is usually irrelevant to students of mathematics unless, of course, they aspire to be mathematicians.

The work of Karl Weierstrass (building on ideas developed by Cauchy) established what many call the soundness of calculus.  He removed, from the limit process, the idea of motion or of changing values of a variable.  The Princeton Companion to Mathematics tells us how he began:

He attempted a unified approach to the definition of rational and irrational numbers which involved unit fractions and decimal expansions and now seems somewhat murky.  Weierstrass’s definition of the real numbers appears unsatisfactory to modern eyes, but the general path of arithmetization of analysis was established by this approach.

Then a bit later:

The former notions of variables tending to given values were replaced by quantified statements about linked inequalities.

These linked inequalities are the epsilon/delta definitions of limits and continuity. In the book, Where Mathematics Comes From (Lakoff/Nunez), which I’ve referenced in other blogs,  the authors try to characterize what Weierstrass had set out to do:

He had to eliminate one of most basic concepts – the natural continuity of a trajectory of motion – and replace it with a concept that involved no motion (just logical conditions), no continuous space (just discrete entities), no points (just numbers), with functions that are not curves in a plane (but, rather, sets of ordered pairs of numbers).

And here, the very thing that I first loved about calculus is being taken away – that the things we were looking at weren’t static and discrete, but seemed to be moving, and often across endless distances.   It was my first calculus class that brought me to attention and opened my eyes to the depth of these creative thoughts.  Of course, this meaning isn’t really taken away and that I find equally provocative. Nunez wrote an interesting paper (Nunez 2008 in his list of publications) about the gestural components of these abstractions.

In another essay in the Princeton Companion to Mathematics, Jose Ferreiros draws attention to the disagreement between Weierstrass and some of contemporaries, namely Riemann, Dedekind, and Cantor, who favored conceptual meaning over calculated or formula driven results. Dedekind’s definition of the real numbers (the Dedekind cut) is judged more successful than Weierstrass’s.  It is grounded in the logic of order, the concept of collections, and the association of numbers with points on a line.  He starts by considering the system of rational numbers as a whole, an actual infinity.  It is a very abstract description of ordered elements which, nonetheless, obey the laws of arithmetic.

While Weierstrass may have been at odds with some of his contemporaries (whose work would have far-reaching consequences), he set something big in motion.  He is credited with bringing the required precision to calculus ideas, grounding them in relationships that could be clearly stated and repeatedly applied.  Newton’s definition of a limit had been given in this way:

Quantities, and the ratios of quantities, which in any finite time converge continually to equality, and before the end of that time approach nearer to each other than by any given difference, become ultimately equal. (Principia Mathematica, Book 1)

And this one from Cauchy was an improvement:

When the values successively attributed to a variable approach indefinitely to a fixed value, in a manner so as to end by differing from it by as little as one wishes, this last is called the limit of all the others. (Cours d’Analyse)

Weierstrass’s definition is strictly arithmetic and gave calculus concepts indisputable validity.  It has been said that Weierstrass’s limit merely hides the more intuitive, geometric and motion-based idea, and that Dedekind’s definition of a real number is embedded in the everyday meanings of cutting and separating things. (See the paper Visualism in Mathematics).

But it is clear that symbol itself opens the door to the perception of more detail (even in its worldly application), which I find most interesting. And maybe here is my real point.  Somehow the mind (or the body) takes its experience of number, which it uses to categorize quantities of things and to compare distances, and extends it.  The experience becomes externalized in symbol. The notation we currently use was found to be best for calculation purposes. Infinite decimal expansions and incommensurable quantities opened other questions, which were addressed from many directions.  And in the 19th century, the simplest of relations (less than or greater than), would provide a completely abstract definition of number, the Dedekind cut, one of a few that now meet the mathematician’s needs.  In the set of real numbers, something new is expressed.

The approach of mathematicians like Dedekind, Riemann and Cantor eventually won out, and confidence in the precision and power of conceptual meaning bore the fruit of modern mathematics.  That confidence comes from an intimacy with the subject, where one can see what is correct and what is not.  What it is that the mathematician is getting closer to is yet to be understood.